Stability and attractors, reply to RKC

CSGnet Archive CSG1993
1

[Martin Taylor 931220 11:30]
(Cliff Joslyn 931219 16:20)

My dynamics book defines an "attractor" as a set of points in a phase
space (e.g. magnitude vs. change of magnitude) such that there exists
am equilibrium point within the attractor, so that if you begin the
system anywhere in the attractor, it will approach the equilibrium in
the limit. The "basin of attraction" is then the largest attractor. So
for the ball in the bowl (with friction), each circular region with
center at the bottom of the bowl is an attractor, the bottom is the
(stable) equilibrium, and the entire bowl is the basin of attraction.

Two points. First of nomenclature: the "attractor" is, in at least some
dynamics books, taken as being the limit toward which orbits in a "basin
of attraction" lead. The limit may be a point, a cycle, or a fractal
structure, but in any case it is the end-point, if such exists, of orbits
in the phase space. The phase space may thus contain many basins of
attraction, which are separated by "repellors" (the boundaries between
two or more basins of attraction). Your book may use different terminology,
but that's what I have been using in writing of attractors and the like.

Secondly: The attractor exists in the phase space, not the physical space.
The centre of the bowl is not an attractor, nor even is it within a single
basin of attraction. The phase space for this situation has two (or three
if you allow the ball to bounce) spatial dimensions and two (or three)
momentum dimensions. The "centre of the bowl" defines a subspace of
constant spatial location extended through the two (or three) dimensional
momentum space. The attractor is at the centre of the bowl at zero
momentum in each dimension of phase space, not just at the centre of the
bowl in physical space.

There is no sense except that of everyday connotation in which an "attractor"
is associated with "an object that attracts." (This for some other posters,
not for Cliff).

Martin

2

Cliff Joslyn 931219 16:20

From Bob Clark (931217.1720 EST)
"Stability" has been suggested as an indicator of the presence of a
negative feedback control system. In many situations, such systems
are "stable," but not necessarily so.

I believe that "stability" is a necessary, but not sufficient,
condition for control. That is, control (good control, I mean)
produces stability, but stability can result from other processes as
well.

A system is said to be "stable" if the disturbed variable tends to
return to its previous condition after the disturbance ceases.
Its stability is "neutral" if it simply continues in accordance with
the effects of the disturbance, even though the disturbance has
ceased.
It is "unstable" if the system acts to further increase the effects
of the disturbance, rather than opposing them.

My dynamics book defines stable, unstable and aymptotically stable in
the context of systems without input. Are your senses above technical,
or are they suggestions as to how we should think of things?

For me, stability is when the "behavior" (meaning dynamical behavior,
the transitions among states) remains in a certain region; unstable is
when it does not; and symptotically stable is when it approaches
arbitrarily close to a region over time.

Introducing the concept of an "attractor" is a distraction at best --
a disturbance to those trying to learn and apply the methods and
concepts of physics.

My dynamics book defines an "attractor" as a set of points in a phase
space (e.g. magnitude vs. change of magnitude) such that there exists
am equilibrium point within the attractor, so that if you begin the
system anywhere in the attractor, it will approach the equilibrium in
the limit. The "basin of attraction" is then the largest attractor. So
for the ball in the bowl (with friction), each circular region with
center at the bottom of the bowl is an attractor, the bottom is the
(stable) equilibrium, and the entire bowl is the basin of attraction.

O----------------------------------------------------------------------------->

Cliff Joslyn, Cybernetician at Large, 327 Spring St #2 Portland ME 04102 USA
Systems Science, SUNY Binghamton NASA Goddard Space Flight Center
cjoslyn@bingsuns.cc.binghamton.edu joslyn@kong.gsfc.nasa.gov

V All the world is biscuit shaped. . .